Calculus Examples

$f (x) = \sqrt{x} + \frac{1}{\sqrt{x}}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \sqrt{x} + \frac{1}{\sqrt{x}}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\sqrt{x} + \frac{1}{\sqrt{x}} = 0$ .

$\sqrt{x} + \frac{1}{\sqrt{x}} = 0$

Step 1.2.2

Solve for $\sqrt{x}$ .

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Step 1.2.2.1

Find the LCD of the terms in the equation.

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Step 1.2.2.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, \sqrt{x}, 1$

Step 1.2.2.1.2

The LCM of one and any expression is the expression.

$\sqrt{x}$

Step 1.2.2.2

Multiply each term in $\sqrt{x} + \frac{1}{\sqrt{x}} = 0$ by $\sqrt{x}$ to eliminate the fractions.

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Step 1.2.2.2.1

Multiply each term in $\sqrt{x} + \frac{1}{\sqrt{x}} = 0$ by $\sqrt{x}$ .

$\sqrt{x} \sqrt{x} + \frac{1}{\sqrt{x}} \sqrt{x} = 0 \sqrt{x}$

Step 1.2.2.2.2

Simplify the left side.

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Step 1.2.2.2.2.1

Simplify each term.

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Step 1.2.2.2.2.1.1

Multiply $\sqrt{x}$ by $\sqrt{x}$ .

${\sqrt{x}}^{2} + \frac{1}{\sqrt{x}} \sqrt{x} = 0 \sqrt{x}$

Step 1.2.2.2.2.1.2

Cancel the common factor of $\sqrt{x}$ .

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Step 1.2.2.2.2.1.2.1

Cancel the common factor.

${\sqrt{x}}^{2} + \frac{1}{\sqrt{x}} \sqrt{x} = 0 \sqrt{x}$

Step 1.2.2.2.2.1.2.2

Rewrite the expression.

${\sqrt{x}}^{2} + 1 = 0 \sqrt{x}$

Step 1.2.2.2.3

Simplify the right side.

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Step 1.2.2.2.3.1

Multiply $0$ by $\sqrt{x}$ .

${\sqrt{x}}^{2} + 1 = 0$

Step 1.2.2.3

Solve the equation.

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Step 1.2.2.3.1

Subtract $1$ from both sides of the equation.

${\sqrt{x}}^{2} = - 1$

Step 1.2.2.3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\sqrt{x} = \pm \sqrt{- 1}$

Step 1.2.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$\sqrt{x} = \pm i$

Step 1.2.2.3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.2.3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$\sqrt{x} = i$

Step 1.2.2.3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$\sqrt{x} = - i$

Step 1.2.2.3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$\sqrt{x} = i, - i$

Step 1.2.3

Solve for $x$ in $\sqrt{x} = i$ .

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Step 1.2.3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x}}^{2} = i^{2}$

Step 1.2.3.2

Simplify each side of the equation.

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Step 1.2.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(x^{\frac{1}{2}})}^{2} = i^{2}$

Step 1.2.3.2.2

Simplify the left side.

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Step 1.2.3.2.2.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 1.2.3.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 1.2.3.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = i^{2}$

Step 1.2.3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = i^{2}$

Step 1.2.3.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = i^{2}$

Step 1.2.3.2.2.1.2

Simplify.

$x = i^{2}$

Step 1.2.3.2.3

Simplify the right side.

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Step 1.2.3.2.3.1

Rewrite $i^{2}$ as $- 1$ .

$x = - 1$

Step 1.2.4

Solve for $x$ in $\sqrt{x} = - i$ .

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Step 1.2.4.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x}}^{2} = {(- i)}^{2}$

Step 1.2.4.2

Simplify each side of the equation.

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Step 1.2.4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(x^{\frac{1}{2}})}^{2} = {(- i)}^{2}$

Step 1.2.4.2.2

Simplify the left side.

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Step 1.2.4.2.2.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 1.2.4.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 1.2.4.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = {(- i)}^{2}$

Step 1.2.4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {(- i)}^{2}$

Step 1.2.4.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = {(- i)}^{2}$

Step 1.2.4.2.2.1.2

Simplify.

$x = {(- i)}^{2}$

Step 1.2.4.2.3

Simplify the right side.

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Step 1.2.4.2.3.1

Simplify ${(- i)}^{2}$ .

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Step 1.2.4.2.3.1.1

Apply the product rule to $- i$ .

$x = {(- 1)}^{2} i^{2}$

Step 1.2.4.2.3.1.2

Raise $- 1$ to the power of $2$ .

$x = 1 i^{2}$

Step 1.2.4.2.3.1.3

Multiply $i^{2}$ by $1$ .

$x = i^{2}$

Step 1.2.4.2.3.1.4

Rewrite $i^{2}$ as $- 1$ .

$x = - 1$

Step 1.2.5

List all of the solutions.

$x = - 1, - 1$

Step 1.2.6

Exclude the solutions that do not make $0 = \sqrt{x} + \frac{1}{\sqrt{x}}$ true.

$No solution$

Step 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \sqrt{0} + \frac{1}{\sqrt{0}}$

Step 2.2

The equation has an undefined fraction.

Undefined

Step 2.3

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $None$

Step 4